Technical report, IDE1066, February 2011 Model of MOSFET in Delphi Master’s Thesis in Microelectronics and Photonics Andrey Prokhorov Olesya Gerzheva School of Information Science, Computer and Electrical Engineering Acknowledgment We want to thank our supervisors: Docent Ying Fu from the Royal institute of Technology for his valuable suggestions which helped us greatly during the process of preparing and writing this thesis; and we also want to thank Doctor Lars Landin who supervised us in the beginning of work. Thanks to Håkan Pettersson, the Head of the Dept. of Mathematics, Physics and Electrical Engineering, and main teacher in semiconductor physics, for his help and advice. Thanks to all members of our families for their support and encouragement. 2 Abstract In modern times the increasing complexity of transistors and their constant decreasing size require more effective techniques to display and interpret the processes that are inside of devices. In this work, we are modeling a two‐dimensional n‐MOSFET with a long channel and uniformly doped substrate. We assume that this device is a large geometry device so that short‐channel and narrow‐width effects can be neglected. As a result of the thesis, a demonstration program was built. In this executable file, the user can choose parameters of the MOSFET‐model: drain and gate voltage, and different geometrical parameters of the device (junction depth and effective channel length). In the advanced regime of the program, the user can also specify the model re‐calculation parameter, doping concentration in n+ and bulk regions. The program shows the channel between the source and drain region with surface diagrams of carrier density and potential energy as an output. It is possible to save all calculated results to a file and process it in any other program, for example, plot graphics in Matlab or Matematica. The model can be used in lectures that are related to semiconductor physics in order to explain the basic working mechanisms of MOSFETs as well as for further detailed analysis of the processes in MOSFETs. It is possible to use our modeling techniques to rebuild the model in another computer language, or even to build other models of transistors, performing similar calculations and approximations. It is possible to download the executable file of the model here: http://studentdevelop.com/projects/MOSFET_model.zip Keywords MOSFET, semiconductor, model, Delphi, channel, carrier concentration, potential energy. 3 Table of Contents Acknowledgment .......................................................................................................................................... 2 Abstract ......................................................................................................................................................... 3 1. Basic theory of semiconductors ................................................................................................................ 5 1.1 Energy bands ....................................................................................................................................... 5 1.2 Intrinsic semiconductor ....................................................................................................................... 5 1.2 Carrier concentration and Fermi level ................................................................................................ 6 2. The MOSFET .............................................................................................................................................. 8 2.1 Introduction......................................................................................................................................... 8 2.2 Energy band diagrams ......................................................................................................................... 9 2.3 The MOSFET ...................................................................................................................................... 12 2.4 Characteristics of the MOSFET .......................................................................................................... 13 2.5 Operating regions of the MOSFET ..................................................................................................... 14 3. Modeling of MOSFET ............................................................................................................................... 17 3.1 Numerical methods ........................................................................................................................... 17 3.2 Description of the model................................................................................................................... 19 3.2.1 Programming model ................................................................................................................... 19 3.2.2 Mathematical model .................................................................................................................. 21 4. Development environment Delphi .......................................................................................................... 23 5. Results ..................................................................................................................................................... 24 6. Conclusion ............................................................................................................................................... 29 References ................................................................................................................................................... 30 Appendix ..................................................................................................................................................... 31 Program code of the model in Delphi ..................................................................................................... 31 Download the model ............................................................................................................................... 35 4 1. Basic theory of semiconductors 1.1 Energy bands Semiconductor materials are the basis of modern electronics. Different kinds of transistors, diodes and solar cells are made of semiconductors. The most popular semiconductor material is silicon. Atoms in a silicon crystal have four valence electrons to share with four nearest neighbors. Electrons of an isolated atom may occupy only certain discrete energy levels. If two atoms in a semiconductor move closer to each other then energy levels split to accommodate all electrons in the system. Usually, the system has a large number of atoms and the higher energy levels tend to unite into two separate bands of allowed energies, called the Conduction band and the Valence band, respectively [1]. The Conduction band – is the upper band, where energy levels are almost empty. Energy level Ec – is the bottom of the conduction band. The Valence band – is the lower band where energy levels are full. Energy level EV – is the top of the valence band. The difference between two of these levels is called the bandgap energy ( Eg )[2]. The bandgap energy of most semiconductors decreases as the temperature increases (Eq.1.1) [1]. ⎧1.206 − 2.73⋅10−4 T (for T ≥ 250 K) ⎪ −5 -7 2 E g (T ) = ⎨1.179 − 9.03⋅10 T - 3.05 ⋅10 T (for 300K > T > 170 K) , (1.1) ⎪ −5 -7 2 ⎩1.17 −1.06 ⋅10 T - 6.05 ⋅10 T (for T ≤ 170 K) where T – is the temperature, (K). 1.2 Intrinsic semiconductor A pure semiconductor without dopant species added is called undoped or intrinsic semiconductor. Intrinsic semiconductor has the same number of electrons in the conduction band as the number of holes in the valence band, at a given temperature. p = n = ni , (1.2) Where p – is the free hole concentration, (cm-3); n – is the free electron concentration, (cm-3); -3 ni – is the intrinsic carrier concentration, (cm ). The formula 1.3 shows the intrinsic carrier concentration as a function of the temperature. 5 T 3/ 2 Eg (T) Eg (T0 ) -3 ni (T) = ni (T0 )( ) exp[− + ] , (cm ) (1.3) T0 2kT 2kT0 where T0 – is the nominal temperature (T0 = 300 K) [1]. 1.2 Carrier concentration and Fermi level If we consider an intrinsic case without impurities added to the semiconductor, then the number of electrons (occupied conduction band levels) is given by the total number of states N(E) multiplied by the occupancyF(E ) , integrated over the conduction band: ∞ ∞ n = ∫ n(E)dE = ∫ N(E)F(E)dE, (1.4) EC EC n(E) – is the electron density [(cm-3-eV)-1], -3 -1 N(E) – is the density of allowed energy states [(cm -eV) ], F(E) – is the Fermi-Dirac distribution function. 3 2mn 2 N(E) = 4π ( 2 ) E − EC (1.5) h where mn – is the effective mass of the electrons; h – is the Planck constant; EC ‐ is the conduction band edge. The Fermi-Dirac distribution function F(E) shows the probability of electron occupation of an electronic state with energy E. 1 F(E) = (E−E ) / kT (1.6) 1+ e F 1 The energy at which the probability of occupation by an electron is called the Fermi 2 energy ( EF ). Large numbers of states are allowed in the conduction and valence band. However, there would not be many electrons in the conduction band for an intrinsic semiconductor because electrons prefer states with lower energy that are in the valence band. Therefore, an electron can occupy one of these upper states with a very small probability. Most of the allowed states in the valence band will be occupied by electrons. Hence, the electron can occupy